Given a graph G = (V, E), a function f: V → {0, 1, 2} is a total Roman {2}-dominating function if every vertex ʋ ∈ V for which f (ʋ) = 0 satisfies that ∑u∈N (ʋ) f (ʋ) ≥ 2, where N (ʋ) repre...

Let G be a graph without isolated vertices. A function $f:V\left(G\right)\to \{0,1,2\}$ is a total Roman dominating function on G if every vertex $v\in V\left(G\right)$ for which $f\left(v\right)=0$ is adjacent to at least one vertex $u\in V\left(G\right)$ such that $f\left(u\right)=2$ , and if the subgraph induced by the set $\{v\in V(G):f(v)\ge 1\}$ has no isolated vertices. The total Roman domination number of G, denoted ${\gamma }_{tR}\left(G\right)$ , is the minimum weight $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ among all total Roman dominating functions f on G. In this article we obtain new tight lower and upper bounds for ${\gamma }_{tR}\left(G\right)$ which improve the well-known bounds $2\gamma \left(G\right)\le {\gamma }_{tR}\left(G\right)\le 3\gamma \left(G\right)$ , where $\gamma \left(G\right)$ represents the classical domination number. In addition, we characterize the graphs that achieve equality in the previous lower bound and we give necessary conditions for the graphs which satisfy the equality in the upper bound above.

Let G be a graph with no isolated vertex and $f:V\left(G\right)\to \{0,1,2\}$ a function. If f satisfies that every vertex in the set $\{v\in V(G):f(v)=0\}$ is adjacent to at least one vertex in the set $\{v\in V(G):f(v)=2\}$, and if the subgraph induced by the set $\{v\in V(G):f(v)\ge 1\}$ has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.

Let G be a graph with no isolated vertex and $f:V\left(G\right)\to \{0,1,2\}$ a function. Let $V_i=\{v\in V\left(G\right):f\left(v\right)=i\}$ for every $i\in \{0,1,2\}$. The function f is an outer-independent Roman dominating function on G if $V_0$ is an independent set and every vertex in $V_0$ is adjacent to at least one vertex in $V_2$. The minimum weight $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}f\left(v\right)$ among all outer-independent Roman dominating functions f on G is the outer-independent Roman domination number of G. This paper is devoted to the study of the outer-independent Roman domination number of a graph, and it is a contribution to the special issue “Theoretical Computer Science and Discrete Mathematics” of Symmetry. In particular, we obtain new tight bounds for this parameter, and some of them improve some well-known results. We also provide closed formulas for the outer-independent Roman domination number of rooted product graphs.

A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$, and - if $x$ is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then $f(x)=1$. The weight of a quasi-total Roman dominating function is the value $\omega(f)=f(V)=\sum_{u\in V} f(u)$. The minimum weight of a quasi-total Roman dominating function on a graph $G$ is called the quasi-total Roman domination number of $G$. We introduce the quasi-total Roman domination number of graphs in this article, and begin the study of its combinatorial and computational properties.